Summary: TORIC RESIDUE CODES:I
ROY JOSHUA AND REZA AKHTAR
Abstract. In this paper, we begin exploring the construction of algebraic codes from toric varieties using toric
residues. Though algebraic codes have been constructed from toric varieties, they have all been evaluation
codes, where one evaluates the sections of a line bundle at a collection of rational points. In the present paper,
instead of evaluating sections of a line bundle at rational points, we compute the residues of differential forms
at these points. We show that this method produces codes that are close to the dual of those produced by the
first technique. We conclude by studying several examples, and also discussing applications of this technique to
the construction of quantum stabilizer codes and also to decryption of toric evaluation codes.
Table of Contents
2. Review of basic techniques
3. Toric residue codes
4. Duality results and estimation of parameters
6. Application I: The construction of quantum stabilizer codes from toric varieties
7. Application II: Decryption of toric evaluation codes
The first author was supported by the IHES, MPI (Bonn) and a grant from the NSA.